Serena Williams has spoken out against police killings of African Americans, writing in a heartfelt Facebook post: “As Dr Martin Luther King said: ‘There comes a time when silence is betrayal.’ I won’t be silent.”

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The tennis champion, arguably one of the greatest sportspeople ever, wrote that she was in a car on Tuesday being driven by her nephew, who is black, when she saw a police car on the side of the road.

“I remembered that horrible video of the woman in the car when a cop shot her boyfriend,” she wrote, referencing Philando Castile, whose girlfriend broadcast on Facebook Live the aftermath of his killing by police. “I even regretted not driving myself. I would never forgive myself if something happened to my nephew. He’s so innocent. So were all ‘the others’.”

“Why did I have to think about this in 2016?” she wrote. “Have we not gone through enough, opened so many doors, impacted billions of lives? But I realized we must stride on – for it’s not how far we have come but how much further still we have to go.”

Williams is the latest – and perhaps the most high-profile – star to join a growing movement of black American athletes who are speaking candidly about how racism and police brutality affect their lives.

In late August, the San Francisco 49ers quarterback Colin Kaepernick set off a nationwide firestorm when he declined to stand for the national anthem to protest about “a country that oppresses black people and people of color”.

Kaepernick initially faced virulent backlash from some sports fans and public figures, but many others showed support. More NFL players have joined the silent, pre-game protest, while others have begun speaking out about racism and police brutality during press conferences that usually focus on sports.

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On Monday, basketball star LeBron James told reporters that he also fears for the lives of his children, saying: “I look at my son being four years removed from driving his own car and being able to leave the house on his own and it’s a scary thought right now to think if my son gets pulled over.”

Williams has spoken in the past about her experience as a black athlete in an overwhelmingly white sport. In 2015, she ended a 14-year boycott of a tournament in Indian Wells, California, where she had been booed and jeered as a 19-year-old.

“The undercurrent of racism was painful, confusing and unfair,” she wrote in an essay for Time. “In a game I loved with all my heart, at one of my most cherished tournaments, I suddenly felt unwelcome, alone and afraid.”

In her Facebook post, however, Williams appeared to promise more vocal activism around racism in policing. “I had to take a look at me,” she wrote. “What about my nephews? What if I have a son and what about my daughters?”